Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in , which are divisi

Question

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]

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(2)

$A = \underset{101 terms}{\underset{⏟}{{1, 6, 11, 16, . . ., 501}}}$

$\Rightarrow T_{n} = 5 n - 4, where 1 \leq n \leq 101$

$B = \underset{71 terms}{\underset{⏟}{{9, 16, 23, . . ., 499}}}$

$\Rightarrow T_{m} = 7 m + 2, where 1 \leq m \leq 71$

For common difference of

$A \cap B = LCM {5, 7} = 35$

$A \cap B = {16, 51, . . .}$

$\Rightarrow T_{K} = 35 K - 19$

$\Rightarrow 16 \leq 35 K - 19 \leq 499$

$\Rightarrow 35 \leq 35 K \leq 518$

$\Rightarrow 1 \leq K \leq 14.8$

$∴$ K has 14 values

Now, terms divided by 3 in $A \cap B$ are {51, 156, 261, 366, 471} [ $∴$ LCM of 35 and 3 is 105]

Hence, the required number of terms is 5.

Solution

Q.
Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]

1 4

2 5

3 6

4 7

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Solution

Q. Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in A∩B, which are divisible by 3, is: [2026]

1 4

2 5

3 6

4 7

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Q.
Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$ , which are divisible by 3, is: [2026]